Ground-state analysis of the Falicov-Kimball model on complete graphs

The ground-state nature of the Falicov-Kimball model of mobile electrons and fixed nuclei on complete graphs is investigated. We give a pedagogic derivation of the eigenvalue problem and present a complete account of the ground-state energy both as a function of the number of electrons and nuclei and as a function of the total number of particles for any value of interaction U(belong to) R We also study the energy gap and show the existence of a phase transition characterized by the absence of gap at the half-filled band for U < 0. The model in consideration was proposedʹ and partially solved by Farkasovsky for finite graphs and repulsive on-site interaction U > 0. Contrary to his proposal, we conveniently scale the hopping matrix to guarantee the existence of the thermodynarnic limit. We also solve this model on bipartite complete graphs and examine how sharp the Kennedy-Lieb variational estimate is as compared with the exact ground state.